The Stein Operator for Exponential-Power Shrinkage: Local Thresholds, Large-Signal Obstruction, and the Exceptional Role of p=2

This paper develops the correct structural framework for radial shrinkage estimation in product exponential-power location models with matched Lp loss, identifying the vector Stein operator as the central object rather than the scalar factor n minus p. For a noise model where each coordinate has a generalized Gaussian distribution with shape parameter p and scale b, the paper derives a vector Stein identity expressing the score in terms of a divergence operator, and uses it to characterize when the radial shrinkage direction x divided by the Lp norm to the power p yields local improvement in risk. The corrected first-variation theorem shows that the directional derivative of risk along the radial field equals negative b to the p times n minus p times the expected inverse p-th power of the Lp norm, which is strictly negative whenever n exceeds p, so the radial direction is locally improving at every parameter point in that range. At the symmetric point where the true parameter is zero, the paper derives exact closed-form expressions for the optimal local step size and the local risk improvement, revealing a key structural fact: although the capacity coefficient is quadratic in n minus p, the realized local gain at the symmetric point is only linear because the pivotal integral collapses by a factor proportional to the reciprocal of n minus p. The paper then conducts large-signal stress tests in the symmetric regime where all coordinates of the true parameter equal t, and the sparse regime where only the first coordinate equals t, both as t grows to infinity. In both cases the exact leading-order asymptotics are computed explicitly. The positive curvature term from the second-order Taylor expansion of the Lp loss decays like t to the power 2 minus 2p, while the negative Stein gain decays like t to the power minus p. For 1 less than p less than 2, the curvature term decays more slowly and eventually dominates, producing a rigorous impossibility theorem: for every fixed step size c greater than zero, the fixed-step radial estimator fails to dominate the maximum likelihood estimator at large signals. An explicit falsification at p equal to 1 using Laplace noise is also proved. The case p equal to 2 is shown to be structurally exceptional: both terms scale identically as t to the power minus 2, and the classical James-Stein coefficient choice makes the combined leading coefficient negative in both regimes. For the subquadratic regime, a taper necessity theorem is proved under explicit regularity conditions: any globally dominating tapered profile must satisfy a prescribed decay rate at infinity, and at the critical tail the large-signal coefficient recovers the James-Stein constant exactly at p equal to 2. A uniform epsilon-approximation theorem is proved for a cutoff profile family, showing that the supremum over all parameter values of the risk excess can be made arbitrarily small by shrinking the cutoff radius, though this is not a domination theorem. For the superquadratic regime p greater than 2, the picture inverts completely. The large-signal obstruction disappears but a singularity at the origin emerges. The correct finite-risk threshold is shown to be n greater than p times p minus 1: when this condition fails the fixed-step rule has infinite risk at the origin; when it holds a global domination theorem is proved for all sufficiently small step sizes by combining a global remainder bound via the Taylor identity with large-parameter decay estimates for the relevant negative-moment integrals. The complete phase diagram for fixed-step radial shrinkage is established across all three regimes. The paper also clarifies the relationship between the Stein operator for exponential-power families and Brown's admissibility program: the Langevin diffusion generated by the Stein operator is a reversible sampling diffusion, not Brown's parameter-space diffusion, so the recurrence-transience admissibility criterion does not directly apply.